normal model hertz

Purpose

This is the default normal contact model that can be used in a wide variety of contexts.

Note

This command is supported by Aspherix GPU.

Syntax

model hertz [other model_type/model_name pairs as described here ] settings keyword values

zero or more keyword/value pairs may be appended after the keyword settings (after all models are specified)

Keywords	Description
limitForce	`on`: ensures that the normal force is never attractive (an artefact that can occur at the end of a collision), `off`: standard implementation that might lead to attractive forces default: `off`
tangential_damping	`on` (activates tangential damping) or `off` default: `on`
heating_normal_hertz	`on`: model contributes to surface heating in the frame of enable_surface_heating, `off`: model does not contributes to surface heating default: `off`
disableNormalWhenBonded	`on`: if the cohesion bond model is used, then the normal force is only added if the two particles are not bonded, `off`: the normal force is always added if two particles overlap default: `off`
computeDissipatedEnergy	`on`: the normal model saves the dissipated energy for each contact for the use in fix calculate/dissipated_energy, `off`: no values are saved default: `off`
wallForceLimiter	`on`: the force that a wall excerts on a particles is limited, `off`: no force limiting is performed default: `off`
wallHardening	`on`: on excessive overlap the force is artificially increased, `off`: no force modification is applied default: `off`

Associated material properties

Material properties

youngsModulus ( $Y$ ): Young’s modulus of the material [pressure]
poissonsRatio ( $\nu$ ): Poisson’s ratio of the material [ $\cdot$ ]

Material interaction properties

coefficientRestitution ( $e$ ): coefficient of restitution of the two materials

Global scalars

hertzOverlapLimit ( $t$ ): relative overlap after which the force is limited (requires wallForceLimiter on) [ $\cdot$ ].
hertzModifiedExponent ( $e$ ): shape of force limiting or hardening curve (requires wallForceLimiter on or wallHardening on) [ $\cdot$ ].
hertzTargetValue ( $t$ ): force multiplier for excessive overlap (requires wallHardening on) [ $\cdot$ ].

Description

This granular model uses the following formula for the normal force between two spherical particles, when the distance $r$ between two particles of radius $r_i$ and $r_j$ is less than their contact distance $d = r_i + r_j$ . There is no force from this model between the particles when $r > d$ :

$F_n = k_n \delta n_{ij} - \gamma_n v_{n,ij},$

where $k_n$ is the spring stiffness, $\delta$ is the overlap of the two particles ( $= d - r$ for spheres), $n_{ij}$ the contact normal, $\gamma_n$ the damping constant and $v_{n,ij}$ the relative normal velocity of the two particles.

In case of non-spherical particles the model is adapted with equivalent definitions. The radius is substituted with the volume equivalent radius and the overlap is defined as the minimum distance between two points on the particles surfaces that lie opposite of each other with respect to the contact point. The latter is the midpoint of the intersection of the two particles.

The spring stiffness for the Hertz model is defined as

$k_n = \frac{4}{3}Y^* \sqrt{r^* \delta_n}, \frac{1}{Y^*} = \frac{1-\nu_i^2}{Y_i} + \frac{1-\nu_j^2}{Y_j}, \frac{1}{r^*} = \frac{1}{r_i} + \frac{1}{r_j},$

where $Y$ and $\nu$ are the Youngs Modulus and Poisson’s ratio, respectively.

The damping constant is given by

$\gamma_n = -2 \sqrt{\frac{5}{6}} \beta \sqrt{S_n m^*}, \beta = \frac{\ln{e}}{\sqrt{ln{e}^2 + \pi^2}}, S_n = 2 Y^* \sqrt{r^* \delta_n}, \frac{1}{m^*} = \frac{1}{m_i} + \frac{1}{m_j},$

where $e$ and $m$ are the coefficient of restitution and particle mass, respectively.

This model contributes to surface heating in the frame of enable_surface_heating if the appropriate flag is activated.

When the cohesion model bond is used the disableNormalWhenBonded keyword can be used. If this parameter is set to on then the normal model will only compute its contribution if the two neighboring particles do not have an active bond. If a bond breaks and the particles overlap the current $\delta_n$ will be set to zero so that no sudden repulsion takes place. This is handled internally by having an offset value that shrinks to zero once the particles start drifting apart.

Force Limiting

Note, that not using limitForce might lead to attractive forces between particles and walls, especially in case the coefficient of restitution is small. Be sure you include this keyword for the pair style and the wall model if you would like to avoid this.

The keywords wallHardening and wallForceLimiter can be used to improve stability for particle flows in which particles can get trapped by moving geometries. Note, the model is only acting on walls and there is no difference to particle-particle contacts.

The wallHardening keyword makes the walls harder when the relative overlap between particles and walls become too large. The force curve can be seen in the image below.

There are two parameters $t$ and $e$ , which can be set using the global scalars hertzTargetValue and hertzModifiedExponent, respectively. The $t$ value determines the maximum force in relation to the normal Hertz model. The second parameter determines the shape of the curve, for larger values the change is less pronounced in the beginning but then more pronounced towards the really high overlap values (compare the blue and green curves above).

The second model can be activated by using the setting wallForceLimiter on in the wall_contact_model command keyword.

The global scalars hertzOverlapLimit and hertzModifiedExponent can be used to change the behaviour of the model. The $t$ parameter (hertzOverlapLimit) determines at what point the limiter starts to become effective whereas the $e$ parameter (hertzModifiedExponent) governs how hard the cut-off is. This model is nearly identical to a force limiter, but does the limiting in a more gradual way without ever becoming fully constant.

Note

This model is actively being developed so use with some caution. If you have any feedback please do not hesitate to contact us as we would love to improve it.

Restrictions

If using SI units, youngsModulus must be > 5e6. If using CGS units, youngsModulus must be > 5e5. When using the limitForce keyword, the specified coefficient of restitution is only approximate. This might become problematic for low coefficients of restitution as shown in Schwager and Poschel.

Coarse-graining information

Using coarsegraining in combination with this command should lead to statistically equivalent dynamics and system state.

Literature

(Di Renzo) Alberto Di Renzo, Francesco Paolo Di Maio, Chemical Engineering Science, 59 (3), p 525-541 (2004). (Note: Wrong definition of G_eq in this paper, corrected in (Di Renzo 2))

(Di Renzo 2) Alberto Di Renzo, Francesco Paolo Di Maio, Chemical Engineering Science, 60 (5), p 1303-1312 (2005).

(Ai) Jun Ai, Jian-Fei Chen, J. Michael Rotter, Jin Y. Ooi, Powder Technology, 206 (3), p 269-282 (2011).

(Brilliantov) Brilliantov, Spahn, Hertzsch, Poschel, Phys Rev E, 53, p 5382-5392 (1996).

(Schwager) Schwager, Poschel, Gran Matt, 9, p 465-469 (2007).

(Silbert) Silbert, Ertas, Grest, Halsey, Levine, Plimpton, Phys Rev E, 64, p 051302 (2001).

(Zhang) Zhang and Makse, Phys Rev E, 72, p 011301 (2005).